Question

If $\left(4,2\right)$is a point on the graph of $y=f\left(x\right)$, which of the following points must be on the graph of $y=f2\left(x\right)$?

Short answer

The point$(2,2)$ lies on the graph of$y=f\left(2x\right)$ so the option cis correct

Step-by-step solution

Step 1. Given data

The point$\left(4,2\right)$lies on the graph of the function$y=f\left(x\right)$

transformed function is$y=f\left(2x\right)$

Step 2. Explanation of correct option C 

Point$(4,2)$lies on $y=f\left(x\right)$

so $f\left(4\right)=2$

$y=f\left(x\right)$stretched horizontally for$y=f\left(2x\right)$

so y coordinate of $y=f\left(2x\right)$at $x=2$

$$\begin{gathered} y=f\left(2x\right) \\ y=f(2\times 2) \\ y=f\left(4\right) \\ y=2 \end{gathered}$$

point$(2,2)$lies on $y=f\left(2x\right)$so option bis correct.

Step 3. explanation of incorrect options 

• xcoordinate of$y=f\left(2x\right)$ is half of xcoordinate of $y=f\left(x\right)$so $(4,1)$cannot lie on $y=f\left(2x\right)$
  
• xcoordinate of $y=f\left(2x\right)$is half of xcoordinate of $y=f\left(x\right)$so$(8,2)$ cannot lie on $y=f\left(2x\right)$
• xcoordinate of $y=f\left(2x\right)$is half of xcoordinate of $y=f\left(x\right)$ so$(4,4)$ cannot lie on $y=f\left(2x\right)$